, Maris Tõnso, Yu KawanoThe paper presents a computation-oriented necessary and sufficient accessibility condition for the set of nonlinear higher-order input-output differential equations. The condition is presented in terms of the greatest common left divisor of two polynomial matrices, associated with the system of input-output equations. The basic difference from the linear case is that the elements of the polynomial matrices belong to a non-commutative polynomial ring. The condition found provides a basis for finding the accessible representation of the set of input-output equations, which is a suitable starting point for the construction of an observable and accessible state space realization. Moreover, the condition allows us to check the transfer equivalence of two nonlinear systems.
1. Aranda-Bricaire, E., Moog, C. H., and Pomet, J.-B. A linear algebraic framework for dynamic feedback linearization. IEEE Trans. Autom. Control, 1995, 40(1), 127–132.
http://dx.doi.org/10.1109/9.362886
2. Bartosiewicz, Z., Kotta, Ü., Pawłuszewicz, E., Tõnso, M., and Wyrwas, M. Reducibility condition for nonlinear discrete-time systems: behavioral approach. Control and Cybernetics, 2013, 42(2), 329–347.
3. Belikov, J., Kotta, Ü., and Tõnso, M. Adjoint polynomial formulas for nonlinear state-space realization. IEEE Trans. Autom. Control, 2014, 59(1).
4. Bell, D. J. and Lu, X. Y. Differential algebraic control theory. IMA J. Math. Control Info., 1992, 9(4), 361–383.
http://dx.doi.org/10.1093/imamci/9.4.361
5. Bronstein, M. and Petkovšek, M. An introduction to pseudo-linear algebra. Theor. Comput. Sci., 1996, 157(1), 3–33.
http://dx.doi.org/10.1016/0304-3975(95)00173-5
6. Conte, G., Moog, C. H., and Perdon, A. Algebraic Methods for Nonlinear Control Systems. Springer-Verlag, London, 2007.
7. Fu, G., Halás, M., Kotta, Ü., and Li, Z. Some remarks on Kähler differentials and ordinary differentials in nonlinear control theory. Syst. Control Lett., 2011, 60(9), 699–703.
http://dx.doi.org/10.1016/j.sysconle.2011.05.006
8. Halás, M. An algebraic framework generalizing the concept of transfer functions to nonlinear systems. Automatica, 2008, 44(5), 1181–1190.
http://dx.doi.org/10.1016/j.automatica.2007.09.008
9. Halás, M. and Kotta, Ü. Transfer functions of discrete-time nonlinear control systems. Proc. Estonian Acad. Sci. Phys. Math., 2007, 56(4), 322–335.
10. Kolchin, E. R. Differential Algebra and Algebraic Groups. Academic Press, 1973.
11. Kotta, Ü., Bartosiewicz, Z., Nõmm, S., and Pawluszewicz, E. Linear input-output equivalence and row reducedness of discrete-time nonlinear systems. IEEE Trans. Autom. Control, 2011, 56(6), 1421–1426.
http://dx.doi.org/10.1109/TAC.2011.2112430
12. Kotta, Ü., Bartosiewicz, Z., Pawłuszewicz, E., and Wyrwas, M. Irreducibility, reduction and transfer equivalence of nonlinear input-output equations on homogeneous time scales. Syst. Control Lett., 2009, 58(9), 646–651.
http://dx.doi.org/10.1016/j.sysconle.2009.04.006
13. Kotta, Ü., Kotta, P., Nõmm, S., and Tõnso, M. Irreducibility conditions for continuous-time multi-input multi-output nonlinear systems. In 2006 9th International Conference on Control, Automation, Robotics and Vision. Singapore, 2006, 807–811.
14. Kotta, Ü., Leibak, A., and Halás, M. Non-commutative determinants in nonlinear control theory: preliminary ideas. In International Conference on Control, Automation, Robotics & Vision. Hanoi, Vietnam, 2008, 815–820.
15. Kotta, Ü., Moog, C. H., and Tõnso, M. The minimal time-varying realization of a nonlinear time-invariant system. In Proceedings of the 9th IFAC Symposium on Nonlinear Control Systems (NOLCOS). France, Toulouse, 2013, 518–523.
16. Kotta, Ü. and Tõnso, M. Realization of discrete-time nonlinear input-output equations: polynomial approach. Automatica, 2012, 48(2), 255–262.
http://dx.doi.org/10.1016/j.automatica.2011.07.010
17. Lu, X. Y. and Bell, D. J. Minimal equivalence of differential io systems. IMA J. Math. Control Info., 1999, 16(2), 165–178.
http://dx.doi.org/10.1093/imamci/16.2.165
18. McConnel, J. C. and Robson, J. C. Noncommutative Noetherian Rings. Birkhäuser, 1987.
19. Messner, W. and Tilbury, D. Control tutorials for MATLAB and simulink, 2011. http://ctms.engin.umich.edu/CTMS/index.php?example=BallBeam\§ion=SystemModeling [accessed 16.10.2013].
20. Ondera, M. The use of Maple in computation of generalized transfer functions for nonlinear systems. In Innovative Algorithms and Techniques in Automation, Industrial Electronics and Telecommunications. Springer, Netherlands, 2007, 275–280.
http://dx.doi.org/10.1007/978-1-4020-6266-7_50
21. Pommaret, J.-F. Partial Differential Control Theory. Vol. II Control Systems. Mathematics and its Applications, Vol. 530, Kluwer Academic Publishers, Dordrecht, 2001.
22. Popov, V. M. Some properties of the control systems with irreducible matrix-transfer functions. Lecture Notes Math., 1969, 144, 169–180.
http://dx.doi.org/10.1007/BFb0059934
23. Sontag, E. From linear to nonlinear: some complexity comparisons. In IEEE Conference on Decision and Control. New Orleans, 1995, 2916–2920.
24. Van der Schaft, A. J. On realization of nonlinear systems described by higher order differential equations. Math. Syst. Theory, 1987, 19(1), 239–275.
http://dx.doi.org/10.1007/BF01704916
25. Van der Schaft, A. J. Transformations of nonlinear systems under external equivalence. In New Trends in Nonlinear Control Theory, LNCIS, 122, 33–43. Springer-Verlag, Berlin, Germany, 1989.
26. Zhang, J., Moog, C. H., and Xia, X. Realization of multivariable nonlinear systems via the approaches of differential forms and differential algebra. Kybernetika, 2010, 46(5), 799–830.
27. Zheng, Y., Willems, J. C., and Zhang, C. A polynomial approach to nonlinear system controllability. IEEE Trans. Autom. Control, 2001, 46, 1782–1788.http://dx.doi.org/10.1109/9.964691
